In this paper, our objective is primarily to use adaptive inverse-quadratic (IQ) and inverse-multi-quadratic (IMQ) radial basis function (RBF) interpolation techniques to develop an enhanced Adam-Bashforth and Adam-Moulton methods. By utilizing a free parameter involved in the radial basis function, the local convergence of the numerical solution is enhanced by making the local truncation error vanish. Consistency and stability analysis is presented along with some numerical results to back up our assertions. The accuracy and rate of convergence of each proposed technique are equal to or better than the original Adam-Bashforth and Adam-Moulton methods by eliminating the local truncation error thus, the proposed adaptive methods are optimal. We conclude that both IQ and IMQ-RBF methods yield an improved order of convergence than classical methods, while the superiority of one method depends on the method and the problem considered.

翻译：本文旨在利用自适应逆二次（IQ）和逆多二次（IMQ）径向基函数（RBF）插值技术，发展改进的Adam-Bashforth与Adam-Moulton方法。通过利用径向基函数中包含的自由参数，使局部截断误差消失，从而增强数值解的局部收敛性。本文给出了一致性与稳定性分析，并结合数值结果支撑我们的论断。通过消除局部截断误差，每种所提方法的精度与收敛阶均达到或优于原始Adam-Bashforth与Adam-Moulton方法，因此所提出的自适应方法具有最优性。我们得出结论：IQ与IMQ-RBF方法均比经典方法具有更优的收敛阶，而方法间的优劣性则取决于具体方法与所考虑的问题。